"Not far from Valley Forge, around the corner from Bravo Pizza, up the road from Paoli Auto Body, there is an odd-looking office building that resembles a stone castle. An eye doctor is on the first floor. On the second floor is a search engine. The proprietor of the search engine is Gabriel Weinberg, who is 33. A few years ago, when Weinberg told his wife about his new business idea - pitting him against more established outfits such as Google and Bing - he admits that she briefly thought he was nuts." What a success story.

They do seem to log at least one thing: the link in the search result you click on.

This is what Google does too, it is to see if what people clicked on to move the more popular higher.

And possibly log other things, probably not very user specific though.

However they could log per IP-address and browser string too which can be very specific.

It's still a big improvement above Google of course:

They don't communicate a referer to the site you visit (because of HTTPS) and they don't have a free analytics service for websites which tracks movement on websites, or own 2 ad-services and or an embedded free video service. And no Facebook-button competitor and no free enbeddable Javascript-libraries and fonts- service for webdevelopers.